Standard +0.3 This is a standard inverse normal problem requiring students to use z-tables to find two z-scores (approximately 0.496 and 1.282), then solve simultaneous equations for μ and σ. It's slightly above routine because it requires setting up and solving the system, but the method is well-practiced in S1 and involves no conceptual difficulty beyond applying the standardization formula twice.
3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.
\(28 - \mu = 0.496\sigma\) (accept 0.495 or in between)
Answer
Marks
Guidance
\(35 - \mu = 1.282\sigma\) (accept 1.281 or in between, but not 1.28)
M1
For any equation with \(\mu\) and \(\sigma\) and a reasonable z value not a prob. Allow ce, \(\sqrt{\sigma}\), \(\sigma^2\), or – and give M1 A0A1ft for these four cases
A1 A1
For 2 correct equations
\(\sigma = 8.91\) (accept 8.89 to 8.92 incl)
Answer
Marks
Guidance
\(\mu = 23.6\)
M1
For solving their two equations by elim 1 variable sensibly
A1
For correct answer
A1
For correct answer
6 marks total
$28 - \mu = 0.496\sigma$ (accept 0.495 or in between)
$35 - \mu = 1.282\sigma$ (accept 1.281 or in between, but not 1.28) | M1 | For any equation with $\mu$ and $\sigma$ and a reasonable z value not a prob. Allow ce, $\sqrt{\sigma}$, $\sigma^2$, or – and give M1 A0A1ft for these four cases
A1 A1 | For 2 correct equations
$\sigma = 8.91$ (accept 8.89 to 8.92 incl)
$\mu = 23.6$ | M1 | For solving their two equations by elim 1 variable sensibly
A1 | For correct answer
A1 | For correct answer
**6 marks total**
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3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.
\hfill \mbox{\textit{CAIE S1 2003 Q3 [6]}}